PC0001

prime_bit_choice_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Construct the zero/one primality indicator at each dense index.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Exact expanded first-order arithmetic statement

forall i. exists e. ((((~(S (i) = 1) /\ forall bpr_left_pc_choice_exists_prime bpr_right_pc_choice_exists_prime. S (i) = bpr_left_pc_choice_exists_prime * bpr_right_pc_choice_exists_prime -> bpr_left_pc_choice_exists_prime = 1 \/ bpr_right_pc_choice_exists_prime = 1)) /\ e = 1) \/ (~((~(S (i) = 1) /\ forall bpr_left_pc_choice_exists_prime bpr_right_pc_choice_exists_prime. S (i) = bpr_left_pc_choice_exists_prime * bpr_right_pc_choice_exists_prime -> bpr_left_pc_choice_exists_prime = 1 \/ bpr_right_pc_choice_exists_prime = 1)) /\ e = 0))

Constructive proof overview

Generated structural guide

Construct the zero/one primality indicator at each dense index.

The unchanged tactic script uses 1 declared prerequisite and contains 15 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_decidable Stable theorem; checked-use authorized

Direct dependents

PC0006 prime_bit_prefix_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

15 script commands · 11 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

01Fix variables and assumptionsL1–1

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro i
02Establish hpL2–4

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime decidable.

  1. L2
    have hp : ((~(S i = 1) /\ forall bpr_left_pc_choice_decision bpr_right_pc_choice_decision. S i = bpr_left_pc_choice_decision * bpr_right_pc_choice_decision -> bpr_left_pc_choice_decision = 1 \/ bpr_right_pc_choice_decision = 1)) \/ ~((~(S i = 1) /\ forall bpr_left_pc_choice_decision_other bpr_right_pc_choice_decision_other. S i = bpr_left_pc_choice_decision_other * bpr_right_pc_choice_decision_other -> bpr_left_pc_choice_decision_other = 1 \/ bpr_right_pc_choice_decision_other = 1))
  2. L3
    specialize prime_decidable (S i)
  3. L4
    apply prime_decidable
03Separate the logical casesL5–5

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L5
    cases hp
04Construct an explicit witnessL6–6

Supply the displayed value, then prove that it has the required property.

  1. L6
    exists 1
05Separate the logical casesL7–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    left
  2. L8
    split
06Use earlier factsL9–9

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L9
    exact hp_left
07Calculate and transport equalitiesL10–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L10
    refl
08Construct an explicit witnessL11–11

Supply the displayed value, then prove that it has the required property.

  1. L11
    exists 0
09Separate the logical casesL12–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L12
    right
  2. L13
    split
10Use earlier factsL14–14

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L14
    exact hp_right
11Calculate and transport equalitiesL15–15

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L15
    refl

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro i
  2. 0002have hp : ((~(S i = 1) /\ forall bpr_left_pc_choice_decision bpr_right_pc_choice_decision. S i = bpr_left_pc_choice_decision * bpr_right_pc_choice_decision -> bpr_left_pc_choice_decision = 1 \/ bpr_right_pc_choice_decision = 1)) \/ ~((~(S i = 1) /\ forall bpr_left_pc_choice_decision_other bpr_right_pc_choice_decision_other. S i = bpr_left_pc_choice_decision_other * bpr_right_pc_choice_decision_other -> bpr_left_pc_choice_decision_other = 1 \/ bpr_right_pc_choice_decision_other = 1))
  3. 0003specialize prime_decidable (S i)
  4. 0004apply prime_decidable
  5. 0005cases hp
  6. 0006exists 1
  7. 0007left
  8. 0008split
  9. 0009exact hp_left
  10. 0010refl
  11. 0011exists 0
  12. 0012right
  13. 0013split
  14. 0014exact hp_right
  15. 0015refl